Improving bounds for value sets of polynomials over finite fields
Jiyou Li, Zhiyao Zhang
TL;DR
The paper improves bounds on the value set size $N_f$ of polynomials over finite fields, sharpening the BSD-type estimate for generic polynomials of degree $d$. By refining the Birch--Swinnerton-Dyer method and exploiting cancellations among Hasse--Weil zeta factors via Artin $L$-functions, it derives the bound $|N_f-\mu_d q| \le {\displaystyle \sum_{r=2}^d \frac{(-1)^r}{r!} (\sum_{\rho} m_{\rho,r}\deg(L(\rho,s)))\sqrt{q}} + C(d)$, where $m_{\rho,r}$ encode the $S_d$-representation structure and $L(\rho,s)$ are Artin $L$-polynomials. In the crucial quartic case ($d=4$) with $p\neq 2,3$, this yields a tight corollary: for generic quartics $f=x^4+ax^2+bx$ there exists a Zariski-open set $U\subset\mathbb{A}^2$ such that $|N_f-\tfrac{5}{8}q|\le \tfrac{1}{2}\sqrt{q}+\tfrac{3}{4}$. The authors also provide a detailed analysis of the combinatorics of $m_{\rho,r}$ via Kostka numbers, and compute the corresponding degrees of the relevant Artin $L$-functions for the quartic symmetric group $S_4$, validating the bound with explicit representations and curve data. This advances understanding of how arithmetic and geometric decompositions interact to bound image sizes in finite fields, with potential extensions to higher degrees.
Abstract
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(α)\midα\in\mathbb{F}_{q}\}$ and denote the cardinality of this set as $N_{f}$. A much sharper bound for $N_{f}$ is established in this paper. In particular, for any $p\neq 2, 3$, and for nearly every generic quartic polynomial $f \in \mathbb{F}_{q}[x]$, we obtain $$\lvert N_f - \frac{5}{8} q \rvert \leq \frac{1}{2}\sqrt{q} + \frac{3}{4},$$ which holds as a simple corollary of the main result.